[FOM] Wittgenstein's analysis on Cantor's diagonal method
chzhuang000 at sina.com
Fri Apr 30 08:22:27 EDT 2010
I have written a paper about Wittgenstein's analysis on Cantor's diagonal method. The abstract is as follows.
In Zettel, Wittgenstein considered a modified version of Cantor’s diagonal argument:
"A variant of Cantor’s diagonal proof:
Let N=F(k, n) be the form of a law for the development of decimal fractions. N is the nth decimal position of the kth development. The law of the diagonal is then: N=F(n, n)=Def. F’(n).
To prove: that F’(n) cannot be one of the rules F’(k, n). Assume it is the 100th. Then the rule for the construction
of F’(1) runs F(1, 1)
of F’(2) F(2, 2) etc.
but the rule for the construction of the 100th position of F’(n) becomes F(100, 100) i.e. it shows only that the 100th place is supposed to be the same as itself, and so for n=100 it is not a rule.
The rule of the game runs ‘Do the same as …’―and in the special case it becomes ‘Do the same as what you do’. "
(Part 694 in Zettel)
According to Wittgenstein, Cantor’s number, different with other numbers, is defined based on a countable set. If Cantor’s number belongs to the countable set, the definition of Cantor’s number become incomplete. Therefore, Cantor’s number is not a number at all in this context. We can see some examples in the form of recursive functions. The definition "f(a)=f(a)" can not decide anything about the value of f(a). The definiton is incomplete. The definition of "f(a)=1+f(a)" can not decide anything about the value of f(a) too. The definiton is incomplete.
According to Wittgenstein, the contradiction, in Cantor's proof, originates from the hidden presumption that the definition of Cantor’s number is complete. The contradiction shows that the definition of Cantor’s number is incomplete.
According to Wittgenstein’s analysis, Cantor’s diagonal argument is invalid. But different with Intuitionistic analysis, Wittgenstein did not reject other parts of classical mathematics. Wittgenstein did not reject definitions using self-reference, but showed that this kind of definitions is incomplete.
Based on Thomson’s diagonal lemma, there is a close relation between a majority of paradoxes and Cantor’s diagonal argument. Therefore, Wittgenstein’s analysis on Cantor’s diagonal argument can be applied to provide a unified solution to paradoxes.
The full paper could be downloaded in the following web address:
Xiamen University, P.R.China
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